6 Low temperature results
6.2 Measurements in non-local geometry
In local geometry the linear shape of the potential profile was unchanged when switching from room temperature to low temperatures. Only the slope that rep-resents the resistivity changed. In non-local geometry, however, the potential profiles look quite different. Also, several measurement runs do not necessarily lie on one general curve as found at room temperature1. In contrast the measured potential fluctuated partially irreproducibly.
At T = 110 K (fig. 6.3) the exponential behavior which was observed unambigu-ously at room temperature (fig. 5.15) disappeares. The potential fluctuates around a mean value of≈3 mV and exhibits a small positive slope. An irregularity occurs between 400 nm and 600 nm where a potential shift can be observed. In a thor-ough examination of the SEM image of the tube a slight anomaly can be noticed at this position (see SEM image in fig. 6.3).
At T = 28 K the potential profile looks similar with two exceptions. First, there is an additional temporary increase in the potential between 100 nm and 200 nm which cannot be attributed to an anomaly in the SEM image. Second, the area
1Although all other measurements consisted also of several individual measurement runs they were plotted in only one color since they were located at one general curve.
- 64 - CHAPTER 6. LOW TEMPERATURE RESULTS
Figure 6.3: Non-local potential profile of sample D at a temperature ofT = 110 K.
The clear exponential behavior at room temperature (see fig. 5.15) cannot be observed any more. In contrast the potential fluctuates with an amplitude up to 1 mV around a nearly constant potential of≈3 mV except in the region between 400 nm and 600 nm where the potential is increased. Around this region there seems to be a small convexity in the SEM image.
Figure 6.4: Non-local potential of sample D at a temperature of T = 28 K. In addition to the potential increase around 500 nm that was already observed at T = 110 K there is a second region with elevated potential between 100 nm and 200 nm. In the region between 200 nm and 400 nm all three measurement runs resulted in the same reproducible oscillations.
CHAPTER 6. LOW TEMPERATURE RESULTS 65 -between the two maxima was reproduced nearly exactly by all three measurement runs. In this region the increasing trend is superimposed by a small reproducible oscillation.
These three effects, namely the irreproducible fluctuations at a small scale (best seen in fig. 6.3), the conspicuouslocal increaseof the potential and thereproducible oscillating potential at 28 K cannot be explained classically. In fact, even in the quantum transport regime an inconstant non-local potential is only possible if nonzero transmission probabilities to both contacts exist (see equation (4.2)), i.e.
even to the electrode farther away. Two pathways to this electrode are imaginable.
First, the straight way passing the second contact in the outermost shell directly below the electrode. Although Ke et al. [126] calculated that the transmission straight through the contact is drastically decreasing even for narrow contacts, Makarovski et al. [127] and Gunnarsson et al. [128] both found a non-constant non-local potential in SWCNTs indicating a non-vanishing transmission through the contact. For the MWCNT used in our experiment, an additional pathway that does not cross the contact is possible. Similar to the room temperature behavior the electrons can exit the outermost shell, pass the second contact in an undis-turbed lower shell and reenter the outermost shell again beyond the electrodes.
Theirreproducible fluctuationsat a small scale for example between 0 and 400 nm at 110 K can be attributed to the limited accuracy of the tip positioning com-bined with interference effects. The digital steps alone aggregate an uncertainty of ≈4 nm. An additional error, which is even more pronounced at low tempera-tures, results from the thermal drift that was corrected only with a drift function linear in time. Furthermore, only the position along the tube axis was recorded.
Since the MWCNT is not strictly one dimensional a variation perpendicular to the tube is also possible. All described effects combine to a uncertainty in the posi-tion of about 10 nm. Consequently small scale effects cannot be recorded with the present technique but can give rise to fluctuations that seems to be irreproducible at first glance.
One possible reason for small scale variations in the potential are induced by Friedel oscillations. They arise from the electron waves in the conductor being reflected at impurities phase coherently. The incoming and reflected waves in-terfere and form standing waves whose anti-nodes (nodes) at position x increase (decrease) the local density of states at point x. Since both contacts inject elec-tron waves that can be handled independently in the phase coherent state, two partial local densities of states exist that are equal to the injectivites of the ac-cording contacts [105]. The oscillations in the injectivities have wave lengths of
- 66 - CHAPTER 6. LOW TEMPERATURE RESULTS
λ = k2πF ≈ 4 ˚A whereby kF denotes the Fermi wavevector. Since the measured potential is the average of the potentials of the electrodes, weighted with the in-jectivities (see equation (4.3)) the resulting potential has also a periodicity in the range of 4 ˚A. This is well below the lateral resolution that results in apparently random oscillations depending on the exact position of the tip on the tube.
Another approach for these fluctuations is given by Makarovski et al. [127] and Gunnarsson et al. [128]. The authors found significant voltages in four-point measurements of SWCNTs in non-local geometry and traced it back to different coupling of the probes to the two orbital modes of the tube. Due to the differ-ent interaction of the two modes with the evaporated contacts, the transmission probabilities can be modulated differently and consequently their electrochemical potentials (as defined in equation (4.3)) differ from each other.
In equation (4.3) it was stated that the measured potential is independent of the coupling strength. This results from the square of the coupling energy contributing linearly to all Tαβ (equation (1) of ref. [105]). In the case of two independently contributing modes, however, the two different coupling constants cannot cancel.
The measured potential depends on the coupling energiestand ˜tto the two modes as follows (˜distinguishes the two modes.):
µ3 =
|t|2νx1+|˜t|2ν˜x1
µ1+|t|2νx2+|˜t|2ν˜x2
µ2
|t|2ν(x) +|˜t|2ν˜(x) (6.3) The measured potentialµ3 lies between the two potentials that could be measured if the probe couples only to one mode.
In the present case of a movable nanocontact the coupling strengths to the two modes can vary and consequently the measured potential deviate from each other.
Due to the limited accuracy in the positioning even if one tries to measure at the same position the coupling constants to the two modes can vary and consequently the potential profile seems to be irreproducible.
Both discussed reasons for the potential fluctuation, the Friedel-like oscillations as well as the differing coupling strengths to the two orbital modes result in principle in well defined potential measurement values1 but occur at such scales that they are not accessible with the available technique. Thus these effects cannot be distinguished with the available data.
1The measuring error extracted from single spectroscopy curves is well below the fluctuation height of≈1 mV.
CHAPTER 6. LOW TEMPERATURE RESULTS 67 -The main effect, in theregions with locally increased potential, occurs at a consid-erably larger scale. Since at least at the region at about 500 nm the structure of the nanotube exhibits a small structural defect a connection seems to be evident.
Obviously this defect does not modify the transport properties considerably at room temperature. Non-local room temperature curves, recorded after the low temperature measurements, show the same exponential behavior as above men-tioned (fig. 5.15) and therefore the differences cannot be attributed to a struc-tural damage that occurred during the low temperature measurements. The room temperature measurements show also that the outer shells cannot be disrupted since this would lead to a horizontal asymptote (see potential profile of sample B, fig. 5.6). In classical terms it is not clear why the measured potential decreases again after the defect to the value before the defect. So, this effect can be only discussed in terms of quantum conductance.
For low temperature it was reported that only outer shells contribute to the trans-port properties [68], consequently the intershell conductance has to be much lower than determined for room temperature. A lower intershell coupling, however, gives rise to the assumption that the electrochemical potentials in the outermost and the second shell are considerably different. If the coupling between the STM tip and the outermost shell of the tube is reduced due to structural defects the feed-back control reduces the tip-nanotube distance in order to keep a constant tunnel current. The direct coupling between tip and second shell is typically negligible due to the higher distance to the tip and the exponential behavior of the tunnel current. With a decreased distance to the tip and a reduced coupling between outermost shell and tip the contribution of the second shell also has to be con-sidered. So the second shell can contribute significantly to the current in the tip resulting in an increased potential.
At the second position where this effect occurs at the lower temperature of 28 K no structural defects at all can be observed within the resolution of the SEM image. Possible reasons for a lower coupling to the outermost shell might be some doping impurities that deplete the local density of states as well as adsorbates on the surface.
The last remarkable effect of the non-local low temperature measurements is the reproducible oscillating potential in the area ranging from 200 nm to 400 nm at T = 28 K. The question why irreproducible fluctuations are strongly suppressed in this area is unsolved so far. Only one outlier in this region could be observed at x = 240 nm. All other values lie close to one general oscillating curve. The explanation for this oscillating potential is, similar to the universal conductance
- 68 - CHAPTER 6. LOW TEMPERATURE RESULTS
Figure 6.5: Several paths of one elec-tron that hits the left scattering cen-ter. At every measurement position (two are illustrated, marked with cir-cles) the injectivity to a probe (tip) is represented by the interference of all possible pathways.
fluctuations, a result of interference of several paths that included phase coherent scattering centers. The illustration in fig. 6.5 shows possible pathways after an electron is scattered at the leftmost scattering center. At the measurement posi-tion all amplitudes of the paths that depend on the scattering cross secposi-tions sum up with appropriate phase relation. Electrons originating from different contact electrodes form a specific interference pattern for each contact whose intensities represents the injectivities at this position. The resulting measured potential can be calculated as in equation 4.3. Depending on the density of scattering centers and on the position of the scattering centers the potential profile exhibit irregular but reproducible oscillations as observed.
In summary, the low temperature potential profile of sample D was measured in local as well as in non-local geometry. The average resistivity increases with decreasing temperature. The resistance increase can be interpreted in terms of weak localization but within the available temperature range and precision it cannot be distinguished between 1D od 2D weak localization.
In the non-local data, however, the exponential behavior that indicates classi-cal diffusive intershell transport disappears at low temperature. Instead of that several quantum conductance effects appear. Fluctuations that appear as irre-producible are a result of small scale potential variations combined with the un-certainty in the lateral position of the probe. Furthermore, regions with strongly increased potential can be attributed to a reduced coupling to the outermost shell and therefore the transmission probability in the second shell can contribute con-siderably. In a region at 28 K where no irreproducible fluctuations occurred an oscillating behavior was observed that can be interpreted in terms of interference of several electron paths similar to universal conductance fluctuations.
7 Summary
In this thesis electronic transport properties of multiwall carbon nanotubes have been examined. In particular, the influence of the multiple shell geometry and the impact of structural defects on the conductance were investigated.
In order to gain access to these material parameters experimentally, MWCNTs were placed on a thin insulator surface covering a conducting layer. Metal elec-trodes with low contact resistance have been evaporated on top as source and drain. The tip of a scanning tunneling microscope was used to localize the tube and to act as a movable voltage probe. For this purpose the spectroscopy mode was used since it allows extracting the potential in a voltage compensated state.
Due to the tunneling contact and the zero current flow the intrinsic properties are not influenced by the voltage probe.
With this experimental setup the potential profile of the outermost shell or the electrode where the tube is covered by metal was recorded as a function of the position while a constant current was flowing through the tube. The contact resistance can be directly read out from the potential step at the metal tube transition. The measurements were performed in a local as well as in a non-local geometry at room temperature and at low temperatures.
The room temperature data were interpreted in terms of classical diffusive trans-port using a resistance model. The present model incorporates two shells where the second represents the effective parameters of the inner tube. It considers the finite dimensions of the contacts and the finite total length of the tube. The model rebuilds well the measured potential profile of a tube without obvious defects and yields additional information on the current paths over the shells and the current density in the injection zone. Deviations between the model and the measured data could be traced back to asymmetrically evaporated contacts. One main re-sult of this work is a considerably higher conductance between the shells at room temperature than published so far [94]. This difference was traced back to a large impact of the array of contacts on the tube of the previous experiments.
The resistance model could also be fitted well to the potential profile of a MWCNT
69
- 70 - CHAPTER 7. SUMMARY
with an incomplete outermost shell. The extracted parameters like intershell conductivity and intrashell resistivity are of the same order of magnitude than those of a tube without obvious defect. Furthermore the potential profile of a tube with an intramolecular junction exhibited an anomaly including a change of slope that was directly attributed to the position where the diameter changes, indicating a transition from a metallic to a semiconducting tube. The local potential as a function of the gate voltage does not reveal any significant gate dependency at room temperature.
One sample, originally without obvious defect, was stretched and kinked by means of applying a force perpendicular to the tube with the STM tip. After manipula-tion, the structure exhibited three regions with individual geometrical properties.
The differences of these regions are manifested in the potential profile with dis-tinct slopes which represent different resistivities. Generally, the resistivities of all regions were increased during manipulation. The sector of the tube protruding the contacts was not manipulated and exhibits an exponentially decreasing potential as the model predicted. The extracted tube parameters are similar to those of the previous samples.
Further experiments on the same sample at low temperatures revealed several temperature dependent effects. The potential profile in local geometry reveals an increasing resistivity with decreasing temperature. It can be attributed to the weak localization theory and therefore indicates a decreasing amount of phase destroying inelastic scattering.
The data recorded in non-local geometry showed several anomalies that have to be described with quantum interference effects. In particular, it can be stated that not only the position of the probe but also the ratio of the coupling strengths to several modes like the orbital modes of one tube or modes of lower shells affect the measured potential. Furthermore, oscillations in the potential profile can be attributed to the interference of several pathways that arise from scattering at inhomogeneously distributed scattering centers.
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